What is Order of Operations Simplifying Expressions?
Kevin gives Don directions to his house: "Go left 3 blocks and then go right 2 blocks." Don wasn't paying close attention. He went 2 blocks to the right and then he went 3 blocks to the left. He followed the directions, but not in the right order. Did he get to Kevin's house?
You have already learned about the commutative property of addition. That is, it doesn't matter in what order you add two quantities: a + b = b + a. The commutative property also holds for multiplication, a x b = b x a.
For some mathematical operations, the order does matter. For example, in subtraction, does a - b = b - a? NO! Using numbers, we see 5 - 3 ≠3 - 5. Subtraction is not commutative! Is division commutative?
When there is only one operation, you start at the left and work towards the right. This may not work with expressions that use more than one operation. Do you remember how to evaluate 1000 + 2 = 7?
In algebra, we use the same operations as in arithmetic (Addition, Subtraction, Multiplication .. etc.), but along with numbers, we also work with letters, or variables.
To evaluate an algebraic expression that contains variables, we follow the same Order of Operations as in arithmetic. You may find some of these expressions look quite complex, but no matter! Following the Order of Operations will yield the correct result.
Before we work out the steps of this algebraic expression, you try it:
12x + 4x + (5 - 3) 2 + 1
Let's use the order of operations to evaluate this expression.
Now, you try this one: x + 3x - 2 + x2
Evaluating algebraic expressions is somewhat like following the directions in a gourmet recipe.
To avoid mistakes, you'll need to read over the entire recipe or expression carefully before deciding how to proceed. Even as you go step by step, you may need to read the entire expression or recipe over again to assure that everything is in order.
You may want to put a grouping symbol around several steps in a recipe to show that they are performed as a unit. Similarly, you use parentheses in algebraic expressions to indicate that all the operations they contain must be considered as a single unit.
Notice that (2 + 3)2 - 1 is NOT the same as 2 + 32 - 1
You can also have parentheses within parentheses. These are called nested parentheses.
Consider this expression: ((13 - 1) + (2 + 4) - x) + 2x
See if you can evaluate it.
Each set of parentheses represents a unit. The inner sets of parentheses are done first, then the outer. In general, for any nested grouping symbols, work from the inside out.